Geology Reference
In-Depth Information
j
¼
D
dc
dx
þ
v
F
c
Thus, defining the single component chemical diffusion coefficient D through
j
¼
D dc
=
dx
ð
Þ;
we have
1
dc
dx
D
¼
D
v
F
c
ð
3
:
33
Þ
This quantity can be identified with the intrinsic diffusion coefficient defined
earlier since the position of a site can be taken as fixed relative to an inert marker
or lattice frame.
A drift force arises from a gradient in the nonideal part of the chemical potential
(note that in the case of ideal mixing there is no drift force, even where there is a
concentration gradient, and the tracer and chemical diffusion coefficients are then
identical, the diffusion being driven entirely by the gradient in entropy of mixing).
A drift force may also arise from an electric field when the diffusing species is
charged, from a gravitational potential, or from the presence of a gradient in
temperature (thermal diffusion), pressure, stress, or concentration of another
component. Taking dc
=
dx to be negative, (
3.33
) indicates that with a sufficiently
large drift force in the backward direction (negative v
F
), D can become negative,
giving ''uphill'' diffusion.
Kinetic theory is thus concerned primarily with calculating the jump frequen-
cies C and the responses v
F
to particular drift forces F. Although quantum-
mechanical tunnelling may exist for light atoms at very low temperatures (Flynn
1972
, Chap. 7), the jump frequency for a particular type of jump can generally be
obtained from transition state theory as
C
¼
me
kT
ð
3
:
34
Þ
where m is the ''attempt'' frequency of the order of atomic vibrational frequencies
(*10
-14
s
-1
) and exp
E
=
k
ð Þ
is a Boltzmann (''success'') factor, E being the
energy barrier to be surmounted, k the Boltzmann constant, and T the absolute
temperature (Lasaga
1981
). When the mechanism requires the formation of a
particular type of defect, E is the sum of the energies of formation and of migration
of this defect. To give a jump frequency of 1 s
-1
, a temperature of about 100 Cis
required if E is 100 kJ mol
-1
(*1 eV atom
-1
), and temperatures of about 500,
1,200, and 2,000 C, respectively, are required for values of E of 200, 400, and
600 kJ mol
-1
.
The effect of the drift force F is to lower the barrier E for forward jumps and to
raise it for backward jumps (Fig.
3.3
). Since the change in the barrier is generally
relatively small (Fd
kT) the effect can be expressed approximately through
(
)
1
1
Fc
kT
dc
dx
D
¼
D
ð
3
:
35
Þ
